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Subgradient-based Lavrentiev regularisation of monotone ill-posed problems

Markus Grasmair, Fredrik Hildrum

TL;DR

The paper develops subgradient-based Lavrentiev regularisation, $\mathcal{A}(u) + \alpha \partial \mathcal{R}(u) \ni f^\delta$, for monotone ill-posed problems in Banach spaces, avoiding adjoint operations and enabling time-causal, real-time computation. It proves a general well-posedness theory and convergence rates under variational source conditions, and demonstrates applicability to TV denoising in convolution-type Volterra equations and to parameter identification in semilinear parabolic PDEs, including practical numerical reformulations. The results provide rigorous error bounds and adaptive strategies for parameter choice, expanding the toolkit for stable reconstruction in monotone operator settings and offering efficient, online-capable algorithms. These contributions have potential impact on real-time data assimilation, online imaging, and inverse problems where causality and regularisation flexibility (beyond quadratic norms) are crucial.

Abstract

We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs.

Subgradient-based Lavrentiev regularisation of monotone ill-posed problems

TL;DR

The paper develops subgradient-based Lavrentiev regularisation, , for monotone ill-posed problems in Banach spaces, avoiding adjoint operations and enabling time-causal, real-time computation. It proves a general well-posedness theory and convergence rates under variational source conditions, and demonstrates applicability to TV denoising in convolution-type Volterra equations and to parameter identification in semilinear parabolic PDEs, including practical numerical reformulations. The results provide rigorous error bounds and adaptive strategies for parameter choice, expanding the toolkit for stable reconstruction in monotone operator settings and offering efficient, online-capable algorithms. These contributions have potential impact on real-time data assimilation, online imaging, and inverse problems where causality and regularisation flexibility (beyond quadratic norms) are crucial.

Abstract

We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators and general regularisation functionals . In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of . It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs.

Paper Structure

This paper contains 20 sections, 15 theorems, 175 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Assumptions
  4. Well-posedness
  5. Convergence rates
  6. Functional-analytic background
  7. Proof of well-posedness
  8. Existence---proof of \ref{['thm:existence']}
  9. Stability---proof of \ref{['thm:stability']}
  10. Convergence---proof of \ref{['thm:convergence']}
  11. Proofs of convergence rates
  12. TV denoising in convolutional Volterra integral problems
  13. Strictly monotone Volterra operators
  14. Description of the subgradient of TV and numerical examples
  15. A parameter-identification problem for a semilinear parabolic PDE
  16. ...and 5 more sections

Key Result

Proposition 2.2

Assume that assumptionsassumptions:space--assumptions:regulariser and assumptions:coercivity is satisfied. Let $u_{\textnormal{ref}} \in \mathbb{X}$ be a fixed reference point and assume that there exists a function ${ \gamma \colon (0, \infty) \to (0, \infty) }$ with superlinear growth, such that the "spherical growth condition" holds and the "directional growth condition" holds for all ${ u

Figures (4)

  • Figure 1: TV denoising for the convolutional Abel operator \ref{['eq:op-Volterra-convolution']}--\ref{['eq:abel-kernel']} with ${ s = 1/3 }$.
  • Figure 2: TV denoising for the Volterra operator \ref{['eq:op-Volterra-convolution']} with exponential kernel ${ x \mapsto \exp(-t/10) }$.
  • Figure 3: TV denoising in the parameter-identification problem \ref{['eq:Lav_parid']} with the choice \ref{['eq:PDE-example']} in \ref{['eq:PDE']}. By ${ y_\alpha^\delta }$ we mean the solution of \ref{['eq:PDE']} corresponding to the reconstruction ${ u_\alpha^\delta = \imath v_\alpha^\delta }$.
  • Figure 4: Observed convergence rates in the ${ \textnormal{L}^2 }$ and ${ \textnormal{L}^1 }$ norms for the parameter-identification problem \ref{['eq:Lav_parid']} with \ref{['eq:PDE-example']} in \ref{['eq:PDE']}.

Theorems & Definitions (30)

  • Proposition 2.2: Alternative growth conditions
  • Theorem 2.3: Existence
  • Theorem 2.4: Stability
  • Theorem 2.5: Convergence
  • Remark 2.6
  • Theorem 2.7: General convergence rates
  • Theorem 2.8
  • Theorem 3.1: Browder--Minty Bro1968a
  • Theorem 3.2: Browder Bro1966a
  • Remark 4.1
  • ...and 20 more